Abstract

We introduce a new algorithm for computing correlations of photon arrival time data acquired in single-molecule fluorescence spectroscopy and fluorescence correlation spectroscopy (FCS). The algorithm is based on rewriting the correlation as a counting operation on photon pairs and can be used with arbitrary bin widths and spacing. The flexibility of the algorithm is demonstrated by use of FCS simulations and single-molecule photon antibunching experiments. Execution speed is comparable to the commonly used multiple-tau correlation technique. Wide bin spacings are possible that allow for real-time software calculation of correlations, even for high count rates.

Figures (3)

Calculating the cross correlation of photon sequences in two channels, A and B (photon times shown as vertical solid lines), by using our new algorithm. The maximum and minimum limits for M bins in time lag τ([τ1min,τ1max),[τ2min,τ2max),…,[τMmin,τMmax)) are added to the arrival time of each photon in channel A. (a) These limits (arrows for τ1min; lighter dotted lines for other limits) for photon 1 from channel A; (b) these limits for photon 2. In this example, M=4, τ1min=0, τ2min=τ1max, τ3min=τ2max, and τ4min=τ3max. To the left of each limit in (a) is index j of the photon in channel B, such that uj>t1+τcurr, where, in turn, τcurr=τ1min, τcurr=τ1max, etc. Similar indices j are shown in (b). The contribution to correlogram Yk for each time-lag bin k is mk−lk. In going from photon 1 to 2 in channel A [comparing (a) and (b)] one makes only small adjustments in the values of lk and mk.

Correlations and fits for simulated data of fluorescent molecules diffusing through a Gaussian detection volume. In this case, c=0.1, and diffusion time τD through the detection volume is 300μs. The correlation is fitted to C(τ)=1+1∕[c(1+τ∕τD)1+τ∕(25τD)]. The correlation calculated with our new algorithm with quasi-logarithmic bin spacings from the multiple-tau correlation algorithm (8 bins per octave, 160 total bins) is shown in black. The dotted light gray curve is a fit with c=0.099±0.002 and τD=294±3μs. The dark gray curve is the correlation with wide bin spacings (2 bins per decade, 12 total bins). A fit (not shown) recovered the values N=0.10±0.01 and 300±100μs. The wider spacing sacrifices some accuracy but increases speed of computation.

Photon antibunching experiments performed on fluorescently labeled DNA oligomers attached to a glass surface. (a) Image of single surface-immobilized DNA oligomers; image size, 20μm×20μm. (b) Long-time-scale correlation over the entire image (bin spacings similar to those in Fig. 2). (c) Short-time scale correlations with linearly spaced bins (25ns bins). The nonnormalized cross correlation for the region with time lags from −1.25to1.25μs is shown [central plot in (c), dotted curve in (b)], along with the regions within 1.25μs of the ±2ms time lags [side plot in (c), solid line in (b)]. The spikes in (c) correspond to pulses from the excitation laser. The solid horizontal line is the average peak height in the region −1.25to1.25μs, and the dotted horizontal line is the average spike height in the regions within 1.25μs of ±2ms time-lag regions. The peak at −0.2μs with low photon pair counts is due to photon antibunching (dark arrow). It is not at 0 time lag owing to a cable delay (150ns) and a software-adjusted digital delay (37.5ns).